Less wrong user titotal has written a new and corrected version of this post, and I suggest that anyone wanting to learn this material should learn it from that post instead.
(In case anyone is curious, the main error in this post is that Eliezer describes the mirror as splitting an incoming state |0⟩ into two outgoing states |1⟩+i|2⟩. However, the overall magnitude of this outgoing state is √2, whereas the incoming state had magnitude 1. This means that the mirror is described by a non-unitary operator, meaning that it doesn’t conserve probability, which is forbidden in quantum mechanics. You can fix this by instead describing the outgoing state as (|1⟩+i|2⟩)/√2.
While it is permitted to do quantum mechanics without normalizing your state (you can get away with just normalizing the probabilities you compute at the end), any operators you apply to your system must still have the correct normalization factors attached to them. Otherwise, you’ll get an incorrect answer. To see this, consider an initial state of |0⟩+|3⟩, where |0⟩ describes the photon heading towards a beam-splitter and |3⟩ described the photon heading in a different direction entirely. This state is unnormalized, which is fine. But if we allow enough time to pass for a photon at |0⟩ to strike the beam-splitter, then according to Eliezer’s description of the beam splitter, we get |1⟩+i|2⟩+|3⟩. This suggests that the probabilities of finding the photon at 1,2,3 will be 13,13,13. But this is incorrect. The correct final state should be (|1⟩+i|2⟩)/√2+|3⟩ and gives probabilities 14,14,12. (I am actually rather embarrassed that I didn’t notice this error myself, and had to wait for titotal to point it out, though in my defence the last time I read the QM sequence was at least 5 years ago, before I really knew much QM.))
I would not characterize that as a version of this post. In particular, it does not share the same underlying philosophical viewpoint and could not be substituted for this post in the context of the original sequence.
Less wrong user titotal has written a new and corrected version of this post, and I suggest that anyone wanting to learn this material should learn it from that post instead.
(In case anyone is curious, the main error in this post is that Eliezer describes the mirror as splitting an incoming state |0⟩ into two outgoing states |1⟩+i|2⟩. However, the overall magnitude of this outgoing state is √2, whereas the incoming state had magnitude 1. This means that the mirror is described by a non-unitary operator, meaning that it doesn’t conserve probability, which is forbidden in quantum mechanics. You can fix this by instead describing the outgoing state as (|1⟩+i|2⟩)/√2.
While it is permitted to do quantum mechanics without normalizing your state (you can get away with just normalizing the probabilities you compute at the end), any operators you apply to your system must still have the correct normalization factors attached to them. Otherwise, you’ll get an incorrect answer. To see this, consider an initial state of |0⟩+|3⟩, where |0⟩ describes the photon heading towards a beam-splitter and |3⟩ described the photon heading in a different direction entirely. This state is unnormalized, which is fine. But if we allow enough time to pass for a photon at |0⟩ to strike the beam-splitter, then according to Eliezer’s description of the beam splitter, we get |1⟩+i|2⟩+|3⟩. This suggests that the probabilities of finding the photon at 1,2,3 will be 13,13,13. But this is incorrect. The correct final state should be (|1⟩+i|2⟩)/√2+|3⟩ and gives probabilities 14,14,12. (I am actually rather embarrassed that I didn’t notice this error myself, and had to wait for titotal to point it out, though in my defence the last time I read the QM sequence was at least 5 years ago, before I really knew much QM.))
I would not characterize that as a version of this post. In particular, it does not share the same underlying philosophical viewpoint and could not be substituted for this post in the context of the original sequence.